Launch Origin and Apparent Acceleration Determination Technique

ABSTRACT

A computer-implemented analysis method is provided for identifying launch position and acceleration of a bogey relative to Earth&#39;s surface based on Earth&#39;s radius. The method includes operations of observing a target position and a velocity vector of the bogey; establishing an altitude of the bogey relative to a tangent line on the Earth&#39;s surface from the target position and the velocity vector; determining a flight path angle of the bogey relative to an intersection line perpendicular to the tangent line; calculating a launch angle of the bogey from the flight path angle and the altitude; evaluating a tilt angle of the bogey from the flight path and launch angles; establishing a distance of the target position to the launch position from the flight path and tilt angles and the Earth&#39;s radius; estimating the acceleration from the distance and a velocity magnitude deriving from the velocity vector; and reporting the acceleration and the launch position.

CROSS REFERENCE TO RELATED APPLICATION

The invention is a Continuation-in-Part, claims priority to and incorporates by reference in its entirety U.S. patent application Ser. No. 13/385,472 filed Feb. 13, 2012 and assigned Navy Case 101122, which is a Continuation-in-Part, claims priority to and incorporates by reference in its entirety U.S. patent application Ser. No. 12/930,168 filed Dec. 8, 2010 and assigned Navy Case 99970, which claims the benefit of priority, pursuant to 35 U.S.C. §119, from provisional Application 61/336,131 filed Jan. 15, 2010.

STATEMENT OF GOVERNMENT INTEREST

The invention described was made in the performance of official duties by one or more employees of the Department of the Navy, and thus, the invention herein may be manufactured, used or licensed by or for the Government of the United States of America for governmental purposes without the payment of any royalties thereon or therefor.

BACKGROUND

The invention relates generally to categorization of aerial bogeys. In particular, the invention relates to techniques to distinguish between ballistic and flat trajectory airborne vehicles and categorize the ballistic vehicles accordingly for appropriate interception response.

An unidentified aerial object or “bogey” represents a prospective threat to a defended asset. To intercept the threat as a target, the defender must categorize ballistic trajectories on the basis of intermittent observation of an object's trajectory. This has represented a difficult problem by conventional analysis techniques using observably available data.

SUMMARY

Conventional techniques for categorizing intermittently observed object trajectories yield disadvantages addressed by various exemplary embodiments of the present invention. In particular, various exemplary embodiments provide a thrust accelerated boost phase associated with the trajectory of an intermittently observed object, such as a bogey presumed to be a missile.

Various exemplary embodiments provide a computer-implemented analysis method for identifying launch position and acceleration of a bogey relative to Earth's surface based on Earth's radius. The operations include observing the bogey's position and velocity vector; establishing the bogey's altitude relative to a tangent line on the Earth's surface; determining the bogey's flight path angle; calculating the bogey's launch and tilt angles; establishing the bogey's distance to the launch position; estimating the acceleration from the distance and a velocity magnitude; and reporting the acceleration and the launch position.

BRIEF DESCRIPTION OF THE DRAWINGS

These and various other features and aspects of various exemplary embodiments will be readily understood with reference to the following detailed description taken in conjunction with the accompanying drawings, in which like or similar numbers are used throughout, and in which:

FIG. 1 is a graphical diagram view of a ballistic trajectory;

FIG. 2 is a graphical diagram view of a thrust-driven trajectory;

FIG. 3 is an elevation vector diagram view of the target relative to launch position;

FIG. 4 is a flowchart identifying the procedural process for launch position determination;

FIG. 5 is a flowchart for determining thrust transience; and

FIG. 6 is a flowchart identifying a procedural process for acceleration calculation.

DETAILED DESCRIPTION

In the following detailed description of exemplary embodiments of the invention, reference is made to the accompanying drawings that form a part hereof, and in which is shown by way of illustration specific exemplary embodi ments in which the invention may be practiced. These embodiments are described in sufficient detail to enable those skilled in the art to practice the invention. Other embodiments may be utilized, and logical, mechanical, and other changes may be made without departing from the spirit or scope of the present invention. The following detailed description is, therefore, not to be taken in a limiting sense, and the scope of the present invention is defined only by the appended claims.

In accordance with a presently preferred embodiment of the present invention, the components, process steps, and/or data structures may be implemented using various types of operating systems, computing platforms, computer programs, and/or general purpose machines.

In addition, those of ordinary skill in the art will readily recognize that devices of a less general purpose nature, such as hardwired devices, or the like, may also be used without departing from the scope and spirit of the inventive concepts disclosed herewith. General purpose machines include devices that execute instruction code. A hardwired device may constitute an application specific integrated circuit (ASIC) or a floating point gate array (FPGA) or other related component.

Conventional techniques for bogey identification rely on additional information beyond observations of the object's trajectory. Conventional techniques include independent indicators that the object's trajectory is accelerating or that the object's altitude is above earth's surface. Such techniques potentially assume that the object's behavior is associated with the object's position. This condition is not always valid, which can lead to interception failure. The instant disclosure describes a technique for categorizing ballistic tracks based on a sequence of system reports of a target's velocity.

FIG. 1 shows a diagram view 100 of a notional ballistic trajectory for an object such as a missile. This view 100 is also presented in application Ser. Nos. 13/385,472 and 12/930,168. For such an object, the overall trajectory can be divided into three phases labeled as BOOST, ASCENT, and DESCENT. Time represents the abscissa 110 and altitude represents the ordinate 120. An inverse parabolic dash line 130 represents the notional trajectory for an exemplary rocket launched missile. While the rocket fires, the missile operates in boost mode (or interval) 140 until motor cutoff 145. The missile continues to climb in altitude due to inertia during the coast ascent mode 150 until reaching apogee 155. Thereafter, gravity overcomes the initial launching force and the missile proceeds downward in altitude during descent mode 160.

For the notional diagram view 100, the ASCENT and DESCENT phases correspond to ballistic motion phases of the trajectory. Observed velocity and acceleration vector quantities have distinct characteristics during each phase:

BOOST—During boost mode 140 until burnout 145 (or alternatively motor cutoff if controllable), acceleration points up, close to the direction of the velocity, which may approach vertical in the upward direction.

ASCENT—During coast 150 (after boost) and through apogee 155, acceleration points down, while velocity gradually changes from up to horizontal.

DESCENT—During descent 160, both quantities point down, with velocity changing from horizontal to potentially near vertical down.

In the diagram view 100, lower arrows 170 on the V-line indicate the velocity direction, while upper arrows 180 on the A-line indicate the acceleration direction. For aircraft trajectories having constant velocity, the acceleration is substantially zero (for all directions, in comparison to a ballistic missile). These considerations enable development of a procedure to distinguish between ballistic and non-ballistic type trajectories.

During the BOOST phase 140, acceleration points up, close to the direction of the velocity, which may be approaching vertical_up. During the BOOST phase a rocket motor provides acceleration force for the object (e.g., a missile). During BOOST phase 140, the net acceleration is the acceleration provided by the rocket motor, minus the acceleration provided by gravity. As understood by artisans of ordinary skill, gravitational acceleration is represented by a vector that, to a reasonable degree of accuracy, points to the center of earth's mass. The ASCENT phase 150 begins after the BOOST phase 140 and continues until the APOGEE state 155 as the missile's inertia continues following motor burnout.

During the ASCENT phase 150, the acceleration points down (because the motor provides no further thrust for acceleration, such that gravity becomes the dominant force affecting the target's motion), while the velocity gradually changes from upward (or upward angularly) to horizontal. At APOGEE 155 (or the highest point in the trajectory in the gravitational field), acceleration points down, while velocity is predominately horizontal. During the DESCENT phase 160, both acceleration and velocity point down, with velocity changing from horizontal to potentially near vertical down. In FIG. 1, the direction of acceleration is indicated with arrows on the line labeled “A”, whereas the direction of the velocity is indicated with arrows on the line labeled “V” as shown.

FIG. 2 shows a diagram view 200 of the notional ballistic trajectory 130 for the missile target with a thrust vector 210 denoting the BOOST phase 140. The thrust vector 210 initiates from a launch position 220. The transient acceleration vectors 180 include a transient thrust component 230 directed upward in the BOOST phase 140 and a constant gravitational component 240 directed downward from earth's inertial mass. This represents the period immediately after launch in which the missile accelerates due to rocket motor thrust until burnout 145. The algorithms incorporated by reference (from application Ser. Nos. 13/385,472 and 12/930,168) provide a conclusion regarding whether the observed target is in the BOOST phase 140 of the trajectory, rather than the ballistic phases: ASCENT (including coast) 150, apogee 155, and DESCENT 160. In contrast to the aforementioned applications, the embodiments described herein pertain to identification of the BOOST phase 140 specifically.

Systems that assume purely ballistic motion assume underlying conditions of unpowered motion to develop particular calculations of interest. The disclosed calculations operate with the underlying assumption that the observed target undergoes a constant powered motion, involving thrust, whether initial or subsequent. Exemplary calculations that follow apply particularly to the BOOST phase 140 of the missile trajectory 130.

Acceleration {right arrow over (a)} can be expressed by the following transient relation:

{right arrow over (a)}(t)={right arrow over (g)}+{right arrow over (T)}(t)  (1)

where {right arrow over (g)} represents gravitational force (continuously downward) operative continuously on the missile target, and {right arrow over (T)} denotes rocket motor thrust (upward) as a function of time t acting to project the missile target into the ballistic trajectory 130. The transient thrust term {right arrow over (T)} has a finite non-zero value during the BOOST phase 140 beginning at launch, and descending to zero at burnout 145 and thereafter in the ballistic portion. Depending on the motor cross-section profile, the chamber pressure, and resulting thrust can be tailored to vary during the burn period after initiation.

FIG. 3 illustrates an elevation view 300 of the assumed relationship between an observed target position used to formulate the disclosed calculations and an extrapolated geometric position of launch origin of that target. That launch position 220 represents an initial location of the target to be engaged, and constitutes its identification as an objective of the disclosed techniques.

A launch coordinate system can be superimposed on the launch position 220 in relation to the curvature of the Earth's surface 305 as a launch horizontal 310 and a launch vertical 315 that passes through the Earth's center 320. Earth's radius R_(e) represents distance along a radius arrow 325 between the Earth's center 320 and the launch position 220. A target bogey denotes its flight position 330 traveling in a direction arrow 335 denoted by velocity vector {right arrow over (v)}. Travel distance D as segment 340 along the thrust arrow 210, denoted by radius vector {right arrow over (r)} between launch position 220 and the target position 330, represent other objectives of the disclosed techniques.

A target coordinate system can be superimposed on the target position 330 as a target horizontal 350 and a target vertical 360 also in relation to the Earth's surface 305 at a tangent location from which altitude distance 370 can be determined as alt. The target coordinate horizontal 350 denotes a tangent line to Earth's surface 305 at a point where, a radial line as the target vertical 360 perpendicular to the tangent between Earth's center 320 and the target position 330 intersects Earth's surface 305. Gravitational acceleration operates perpendicular to the surface 305 and thus along the launch and target verticals 315 and 360. The angle 90° minus the flight path angle A_(FP) as arc 385 that defines the target's velocity vector {right arrow over (v)} as arrow 335 from the target vertical 360.

Three angles provide information for obtaining the objectives: flight path angle π/2−A_(FP) denoted by arc 385, launch angle A_(L) denoted by arc 380 and tilt angle A_(T) denoted by arc 390. For simplification, the target 330 and launch position 220 can be treated within two-dimensional planar Euclidean geometry to form a triangle defined by the launch position 220, the target 330 and the Earth's center 320. This arrangement permits the assumption that vector triangles total to 180° or π radians.

Based on this information of Earth's radius R_(e) as arrow 325, altitude alt as distance 370 and flight path angle A_(FP) as arc 385, trigonometric relations can ascertain the initial launch angle A_(L) as arc 380, tilt angle A_(T) as arc 390 between launch position 220 and target position 330, and distance D as segment 340 from launch position 220 to target position 330. For example, a Euclidean triangle with side lengths s_(a), s_(b), and s_(c) and interior angles θ_(a), θ_(b) and θ_(c) can be related by the expression:

$\begin{matrix} {\frac{s_{a}}{\sin \; \theta_{a}} = {\frac{s_{b}}{\sin \; \theta_{\; b}} = {\frac{s_{c}}{\sin \; \theta_{c}}.}}} & (2) \end{matrix}$

For the target boost geometry, the points to the triangle constitute Earth's center 320, the launch position 220 and the target 330.

The angles formed by this target boost geometry at the vertices formed by the triangle points respectively represent the tilt angle A_(T), the launch angle A_(L) plus 90° (or A_(L)+π/2), and 90° minus the flight path angle A_(FP) (or π2−A_(FP)). The sides opposite to these respective angles represent the distance D, Earth's radius plus altitude R_(e)+alt, and Earth's radius R_(e). As described in application Ser. No. 13/385,472, both the altitude alt and the flight path angle A_(FP) can be developed from the target position 330 as given in the observer's local horizontal coordinate system.

The proportionality of these angles and their opposite sides can be expressed (in radians) as:

$\begin{matrix} {\frac{R_{e}}{\sin \left( {{\pi/2} - A_{FP}} \right)} = {\frac{R_{e} + {alt}}{\sin \; \left( {A_{L} + {\pi/2}} \right)} = {\frac{D}{\sin \; A_{T}}.}}} & (3) \end{matrix}$

Mean sphere Earth's radius R_(e) as arrow 325 is established as 6371 km. Target altitude alt and flight path angle A_(FP) can be measured from an observer's perspective. Application Ser. No. 13/385,472 features eqn. (1) for determining the flight path angle A_(FP) (denoted there as FPA). From eqn. (3), the congruence equality yields:

$\begin{matrix} {{{R_{e}{\sin \left( {A_{L} + \frac{\pi}{2}} \right)}} = {\left( {R_{e} + {alt}} \right){\sin \left( {\frac{\pi}{2} - A_{FP}} \right)}}},} & (4) \end{matrix}$

which can be expressed as:

$\begin{matrix} {{{\cos \left( A_{L} \right)} = {\left( \frac{R_{e} + {alt}}{R_{e}} \right){\cos \left( A_{FP} \right)}}},} & (5) \end{matrix}$

the launch angle A_(L) can be determined as:

$\begin{matrix} {A_{L} = {{\arccos \left( {\frac{R_{e} + {alt}}{R_{e}}{\cos \left( A_{FP} \right)}} \right)}.}} & (6) \end{matrix}$

The interior angles A_(sum) of a Euclidean triangle must sum to 180° (or π radians) as:

$\begin{matrix} {{A_{sum} = {\pi = {A_{T} + \left( {\frac{\pi}{2} + A_{L}} \right) + \left( {\frac{\pi}{2} - A_{FP}} \right)}}},} & (7) \end{matrix}$

where all angles are expressed in radians, such that π=180° and π/2=90°. Consequently, the tilt angle A_(T) can be determined as:

A _(T)=π−(½−A _(FP) +A _(L)+½π)=A _(FP) −A _(L).  (8)

From these relations, the travel distance 340 to the launch position 220 can be determined. In particular, the distance D can be determined from eqn. (3) by:

$\begin{matrix} {D = {\frac{R_{e}\sin \; A_{T}}{\sin \left( {{\pi/2} - A_{FP}} \right)} = {\frac{R_{e}\sin \; A_{T}}{\cos \; A_{FP}}.}}} & (9) \end{matrix}$

Apparent acceleration a_(app) of the target during BOOST phase 140 can be determined based on distance D and time t from launch. Integrating distance from acceleration provides:

D=x ₀+1/2vt=x ₀ +v ₀ t+½a _(pp) t ²,  (10)

where initial distance and velocity at the launch position 220 are both zero (x₀=0, v₀=0). Velocity magnitude v of the target in flight can be determined from eqn. (12) in application Ser. No. 12/930,168 and from:

v=v ₀ +a _(app) t  (11)

Thus, time of flight from launch corresponds to:

$\begin{matrix} {t = {\frac{v}{a_{app}} = {\frac{2D}{v}.}}} & (12) \end{matrix}$

Thus, apparent acceleration can be determined by:

$\begin{matrix} {{a_{app} = {\frac{v}{t} = \frac{v^{2}}{2D}}},} & (13) \end{matrix}$

so that acceleration depends on the square of the velocity magnitude. These relations enable acceleration of the target as measured by the observer to be calculated based on previously established parameters, such as two of distance from the launch position 220, velocity and time of flight. Such information can be supplied to an interceptor or alternate fire control system to engage the bogey.

Such information as distance, time and acceleration enables defenders to counterattack as decided. This thrust scenario immediately after launch until booster motor cutoff constitutes an initial BOOST phase. Such a scenario at the launch position 220 assumes that initial velocity and altitude from Earth's surface 305 would both be zero. Subsequent acceleration from post-boost thrust contribution by a secondary rocket motor stage represents a subsequent BOOST phase. For subsequent thrust scenarios, the launch position would be replaced by a previous coast condition, with non-zero velocity and altitude.

The choice of coordinate system is important when performing the initial calculation of flight path angle and altitude. Upon determination, the altitude and flight path angle constitute physical quantities that are equivalent in both an observer's local horizontal coordinate system and a target local horizontal coordinate system, and even in a launch position local horizontal coordinate system.

The launch point position 220 can be described as radially distant from the earth's center by displacement radius R_(e), depicted by arrow 325, to the surface 305. Subsequent from launch, the target (or other object of interest) can be located at position 330. For simplification purposes of this formulation, the target is assumed to travel along its velocity vector {right arrow over (v)} as arrow 335 from the launch position 220 (or alternate mid-flight initializing position) to the observed target position 330.

This subsequent target position 330 can be described by the Earth's radius R_(e) plus altitude alt. The target position 330 is angularly displaced from the displacement radius R_(e) by the tilt angle A_(T). The elevation distance of the target's position 330 from Earth's surface 305 can be expressed as altitude vector 370 from the surface 305. The remote altitude vector 370 is approximately collinear to the target vertical 360, passing between the earth's center and the target position 330, and offset from the Earth's radius R_(e) by the tilt angle A_(T).

The process for determining launch origin parameters include the following operations:

(a) calculate flight path angle A_(FP) of velocity vector 335 with respect to target horizontal 350, and altitude alt of the target above target horizontal 350; (b) determine launch angle A_(L) by the Euclidean triangle identity in eqn. (4) via Earth's radius R_(e), flight path angle A_(FP) and altitude alt; (c) determine tilt angle A_(T) from the interior angles in eqn. (8); (d) establish distance D from current target position to launch position from the triangle identity in eqn. (9); (e) estimate time since launch from eqn. (12) based on distance D, velocity magnitude v and values of initial velocity and position being both set to zero for launch; and (f) determine constant acceleration value from eqn. (13).

Formulae can be sought for an initial value of the launch position 220 derived from the velocity vector {right arrow over (v)} with respect to the target horizontal 350 to the spheroid surface 305 at the target position 330. Components of the velocity can be solved for the target horizontal coordinate system (THCS), labeled as Target Horizontal. The following parameters can be defined:

{right arrow over (r)} coordinates of position vector 210 for the target position 330, e.g., in observer frame coordinates.

{right arrow over (v)} coordinates for velocity vector 335 for the target position 330, in observer frame coordinates.

alt altitude 370 above reference spheroid surface 305 (e.g., Earth's) at the target position 330.

The position vector {right arrow over (r)} is provided in observer frame coordinates: horizontal 310 and vertical 320 as (r_(x) r_(y) r_(Z))^(T), in which superscript T denotes matrix transpose. Similarly, the velocity vector {right arrow over (v)} is provided in observer frame coordinates as (v_(x) v_(y) v_(Z))^(T). A unit vector in the direction of the velocity vector can be calculated from the velocity vector using the normalized vector equation

${{\overset{\rightarrow}{e}}_{v} = \frac{\overset{\rightarrow}{v}}{\overset{\rightarrow}{v}}},$

such that |{right arrow over (v)}| denotes the absolute magnitude of the velocity vector. The distance D from the target's observed position 330 to its launch position 220 is from eqn. (9) is used here with {right arrow over (r)}, and the unit vector in direction of velocity {right arrow over (e)}_(v) to calculate the initial value for launch position 220 in the same observer frame coordinates, given as:

LP={right arrow over (r)}−D·{right arrow over (e)} _(v),  (14)

such that launch position LP represents a position on Earth's surface 305 as a difference between the launch vector and the distance in the velocity vector direction. Further well known modifications to this initial calculation are needed to account for earth motion (change in observer frame coordinates) during the time since launch, e.g., eqn. (12) The modification involves conversion between observer frame coordinates (commonly called ENU) and Earth Centered Earth Fixed (ECEF) coordinates, with compensation for earth rotation at fixed rate.

FIG. 4 shows a flowchart diagram 400 of a bogey identification process for designation as a ballistic missile target (analogous to FIG. 10 from application Ser. No. 13/385,472). A noisy signal 405, such as radar return energy from the bogey, reaches a sensor 410 that detects the signal 405. The sensor 410 submits that information to a preprocessing filter 420 for estimating the observed bogey state, including its position and velocity. The altitude of the bogey can also be determined from its relative position in relation to earth's radius.

A coordinate query 430 can be performed on the resulting estimated state as to whether that state is provided in the target horizontal coordinate system (THCS). If FALSE (as generally expected), the process diverts to a conversion operation 440 using the coordinate transformations described in application Ser. No. 12/930,168. Then, or otherwise if the query is TRUE, the process proceeds to a signal quality query 450 as to whether the filtered and transformed state has an adequate signal-to-noise (SNR) ratio. If not, the process determines at operation 460 that the bogey's state is indeterminate, and that further informtion may be required. Noisy signals compromise the ability to determine acceleration, unless velocity exceeds some SNR threshold, such as for a ballistic missile.

If SNR is adequate, the process diverts to a calculation operation 470 to determine flight path and acceleration path angles (A_(FP) and A_(AP)) respectively from the prior applications. A set of trajectory decision queries 480 uses the calculated path angles to categorize phase states 490 based on the signal 405. In particular, the first query 480 a inquires whether absolute accelera tion exceeds twice gravity (|a|>2g), and both acceleration path and flight path angles greatly exceed zero (A_(AP)>>0°, A_(FP)>>0°), which if all satisfied determine that the bogie can be identified as a ballistic missile with the BOOST phase identifier 490 a. Otherwise, the second query 480 b inquires whether acceleration path angle points vertically downward (A_(AP)˜−π/2=−90°), flight path angle is significantly greater than zero (A_(FP)>>0°), and altitude is above commercial flights, which if all satisfied determine that the bogie can be identified as a missile with the ASCENT phase identifier 490 b.

Otherwise, the third query 480 c inquires whether absolute acceleration approximates gravity (|a|˜g), acceleration path angle points vertically downward (A_(AP)˜−π/2=−90°), and absolute flight path angle approximates zero (|A_(FP)|˜0°), which if all satisfied determine that the bogie can be identified as a missile with the APOGEE state identifier 490 c. Otherwise, the fourth query 480 d inquires whether absolute acceleration approximates gravity (|a|˜g), acceleration path angle points vertically downward (A_(AP)˜−π/2=−90°), and flight path angle is approximately down (A_(FP)˜<0°), which if all satisfied determine that the bogey can be identified as a missile with the DESCENT identifier 490 d. For conditions in which none of these criteria are satisfied, the process concludes at determination 495, such that the bogey constitutes a possible aircraft or other aerial object, that identification is indeterminate, beyond not showing attributes of a ballistic missile. Such negative determination indicates the bogey requires additional analysis prior to categorization as a ballistic missile. Each path 490 a, 490 b, 490 c, and 490 d all follow-up with transition to other processing of block 495.

FIG. 5 shows a flowchart diagram 500 of a BOOST indicator process in relation to the query 480 a to determine the BOOST phase identifier 490 a. A first query 510 inquires whether the ballistic indicator is in BOOST phase 140. If positive, a second query 520 inquires whether the condition is an initial BoosT phase 140. If the first query 510 is negative, the process performs a launch point calculation 530, and a launch time calculation 540. If the second query 520 is negative, the process performs an acceleration calculation 550. Otherwise, if the second query 520 is positive, the process performs a launch time calculation 560 followed by a launch point calculation 570 and an acceleration calculation 580. Following one of the calculations 540, 550 or 580, the process continues 590 to other processing 495.

The acceleration calculation 550 corresponds to indices start-to-boost (S2B) and end-of-boost (E2B), defining parameters time t, distance vector {right arrow over (r)} and velocity vector {right arrow over (v)}:

t ₀ =t _(S2B),

t _(f) =t _(E2B),

{right arrow over (x)} ₀ ={right arrow over (x)} _(S2B),

{right arrow over (x)} _(f) ={right arrow over (x)} _(E2B),  (15)

{right arrow over (v)} ₀ ={right arrow over (v)} _(S2B), and

{right arrow over (v)} _(f) ={right arrow over (v)} _(E2B).

The acceleration quantity of interest can be calculated by the absolute difference relation:

$\begin{matrix} {{a = \frac{2\left\lbrack {\frac{{{\overset{\rightarrow}{x}}_{f} - {\overset{\rightarrow}{x}}_{0}}}{{t_{f} - t_{0}}} - {{\overset{\rightarrow}{v}}_{0}}} \right\rbrack}{{t_{f} - t_{0}}}},} & (16) \end{matrix}$

or if the velocity stream is reliable, from the relation:

$\begin{matrix} {a = {\frac{2\left( {{{\overset{\rightarrow}{v}}_{f}} - {{\overset{\rightarrow}{v}}_{0}}} \right)}{{t_{f} - t_{0}}}.}} & (17) \end{matrix}$

The distinction between a BOOST phase and a post-BOOST condition constitutes the presence or absence of vertical acceleration upward to overcome gravity. The distinction between an initial BOOST phase and a subsequent BOOST phase can be summarized as absence or presence of a prior phase indication after passing the BOOST query. The initial BOOST phase corresponds to the initial observation period in which there is no phase indication other than BOOST from FIG. 4, such as from absence of prior information. A subsequent BOOST phase occurs when ASCENT, APOGEE, or DESCENT phases has been indicated after a prior BOOST observation period. Such indication can be determined from position and velocity vector measurements.

FIG. 6 shows a flowchart diagram 600 of a calculation process as described for determining acceleration from which to determine BOOST phase 140. A first operation 610 calculates flight path angle A_(FP), velocity magnitude |{right arrow over (v)}| and altitude alt. A second operation 620 calculates launch angle A_(L) according to eqn. (6). A third operation 630 calculates tilt angle A_(T) according to eqn. (8). A fourth operation 640 calculates distance D according to eqn. (9). A fifth operation calculates acceleration from velocity according to eqn. (13). A sixth operation calculates launch position 220 from distance and observed target position 330 according to eqn. (14).

These operations, and portions thereof, can be performed as procedural steps being performable by machine operation—e.g., a programmable code for a general purpose computer, or else an ASIC. The process results in an automated determination, based on noisy input data and optional transforma tions, with an identity determination that, a bogey constitutes a ballistic threat.

The following information describes an algorithmic process for calculating the velocity and acceleration directions of a bogey, referred to herein as a “target” generally. The process starts with an observed (or calculated) velocity data stream. The process then tests those values to determine transition from BOOST phase to ASCENT intervals for purpose of categorizing the velocity data stream as representative of a ballistic-type trajectory. The analysis assumes that input velocity stream is or can be represented in the coordinate system horizontal to Earth surface 305 at the target's position 330.

The process for calculating the distance from observed target position 330 to the launch position 220, launch time, launch acceleration, and launch position 220 can be described starting with an observed (or calculated) position and velocity. The process for determining subsequent acceleration and ballistic categorization described in previous disclosures can be based upon observed velocity data stream. The computations are performed in a general purpose programmable computer having instructions recorded on a machine-readable medium. The system includes an observation apparatus, such as an optical observation system, an infrared observation system, or a radar based observation system and the like, which can monitor and observe the trajectory of an object as a function of increments i of time t_(i). Information that can be observed includes position as a function of time, and velocity as a function of time.

These operations, and portions thereof, can be performed as procedural steps being performable by machine operation—e.g., a programmable code for a general purpose computer, or else an ASIC. The process results in an automated determination, (based on noisy input data, optional transformations, and an identity determination that a bogey constitutes a ballistic threat) of the distance from launch, time since launch, launch point, and launch acceleration.

The objective of this technology is to provide alternative methods for calculating certain functions routinely calculated by ballistic missile defense systems, particularly identification of the portion immediately following launch in which the target missile experiences positive thrust. Parent application (Ser. Nos. 12/930,168 and 13/385,472) document development of simple algorithms to distinguish between the various phases of ballistic trajectories. This disclosure documents development of alternative calculations for the case in which the ballistic phase is determined to be under positive thrust with which to determine the bogey's launch origin and its flight acceleration, in order to engage the bogey for interception as well as provide positional information with which to disable subsequent launches through retaliation.

While certain features of the embodiments of the invention have been illustrated as described herein, many modifications, substitutions, changes and equivalents will now occur to those skilled in the art. It is, therefore, to be understood that the appended claims are intended to cover all such modifications and changes as fall within the true spirit of the embodiments. 

What is claimed is:
 1. A computer-implemented analysis method for identifying launch position and acceleration of a bogey relative to Earth's surface based on Earth's radius, said method comprising: observing a target position and a velocity vector of the bogey; establishing an altitude of the bogey relative to a tangent line on the Earth's surface from said target position and said velocity vector; determining a flight path angle of the bogey relative to an intersection line perpendicular to said tangent line; calculating a launch angle of the bogey from said flight path angle and said altitude; evaluating a tilt angle of the bogey from said flight path and launch angles; establishing a distance of said target position to the launch position from said flight path and tilt angles and the Earth's radius; estimating the acceleration from said distance and a velocity magnitude deriving from said velocity vector; and reporting said acceleration and said launch position.
 2. The method according to claim 1, wherein said launch angle can be determined by cosine of the flight path angle multiplied by arc-cosine of a normalized altitude, such that said normalized altitude constitutes a ratio of a difference between the Earth's radius plus said altitude divided by the Earth's radius.
 3. The method according to claim 1, wherein said tilt angle can be determined by said launch angle minus said flight path angle.
 4. The method according to claim 1, wherein said distance can be determined by the Earth's radius multiplied by sine of said tilt angle divided by cosine of said flight path angle.
 5. The method according to claim 1, wherein said acceleration can be determined by square of said velocity magnitude divided by twice said distance.
 6. The method according to claim 1, wherein said acceleration assumes zero initial values of initial position and initial velocity of the bogey.
 7. The method according to claim 1, wherein a time from launch can be determined by twice said distance divided by said velocity magnitude. 